cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A123747 Numerators of partial sums of a series for sqrt(5).

Original entry on oeis.org

1, 7, 41, 9, 239, 6227, 32059, 163727, 166301, 841229, 21215481, 106782837, 536618341, 538698461, 172897, 13538601629, 67813224223, 339532842359, 339895847771, 1700893049407, 42549895540939, 212857129279583, 1064706466190659, 1065035803419763, 5326468921246139
Offset: 0

Views

Author

Wolfdieter Lang, Nov 10 2006

Keywords

Comments

Denominators are given by A123748.
The sum over central binomial coefficients scaled by powers of 5, r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k, has the limit lim_{n -> infinity} r(n) = sqrt(5). From the expansion of 1/sqrt(1-x) for x=4/5.

Examples

			a(3) = 9 because r(3) = 1+2/5+6/25+4/25 = 9/5 = a(3)/A123748(3).

Links

Crossrefs

Cf. A001077/A001076 continued fraction convergents for sqrt(5).
Cf. A123748, A123749.

Programs

  • GAP
    List([0..25], n-> NumeratorRat(Sum([0..n], k-> Binomial(2*k,k)/5^k )) ); # G. C. Greubel, Aug 10 2019
  • Magma
    [Numerator( (&+[Binomial(2*k,k)/5^k: k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 10 2019
  • Maple
    A123747:=n-> numer(sum(binomial(2*k,k)/5^k, k=0..n)); seq(A123747(n), n=0..25); # G. C. Greubel, Aug 10 2019
  • Mathematica
    Table[Numerator[Sum[Binomial[2*k, k]/5^k, {k,0,n}]], {n, 0, 25}] (* G. C. Greubel, Aug 10 2019 *)
  • PARI
    vector(25, n, n--; numerator(sum(k=0,n, binomial(2*k,k)/5^k))) \\ G. C. Greubel, Aug 10 2019
  • Sage
    [numerator( sum(binomial(2*k,k)/5^k for k in (0..n)) ) for n in (0..25)] # G. C. Greubel, Aug 10 2019

Formula

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k, in lowest terms.
r(n) = Sum_{k=0..n} ((2*k-1)!!/((2*k)!!))*(4/5)^k, n>=0, with the double factorials A001147 and A000165.

A123748 Denominators of partial sums of a series for sqrt(5).

Original entry on oeis.org

1, 5, 25, 5, 125, 3125, 15625, 78125, 78125, 390625, 9765625, 48828125, 244140625, 244140625, 78125, 6103515625, 30517578125, 152587890625, 152587890625, 762939453125, 19073486328125, 95367431640625, 476837158203125, 476837158203125, 2384185791015625
Offset: 0

Views

Author

Wolfdieter Lang, Nov 10 2006

Keywords

Comments

Denominators of sums over central binomial coefficients scaled by powers of 5.
Numerators are given by A123747.
For the rationals r(n) see the W. Lang link under A123747.

Examples

			a(3) = 5 because r(3) = 1+2/5+6/25+4/25 = 9/5 = A123747(3)/a(3).

Links

Crossrefs

Cf. A123747, A123749.

Programs

  • GAP
    List([0..25], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k,k)/5^k )) ); # G. C. Greubel, Aug 10 2019
  • Magma
    [Denominator( (&+[Binomial(2*k,k)/5^k: k in [0..n]])): n in [0..25]]; // G. C. Greubel, Aug 10 2019
  • Maple
    A123748:=n-> denom(sum(binomial(2*k,k)/5^k, k=0..n)); seq(A123748(n), n=0..25); # G. C. Greubel, Aug 10 2019
  • Mathematica
    Table[Denominator[Sum[Binomial[2*k, k]/5^k, {k,0,n}]], {n, 0, 25}] (* G. C. Greubel, Aug 10 2019 *)
  • PARI
    vector(25, n, n--; denominator(sum(k=0,n, binomial(2*k,k)/5^k))) \\ G. C. Greubel, Aug 10 2019
  • Sage
    [denominator( sum(binomial(2*k,k)/5^k for k in (0..n)) ) for n in (0..25)] # G. C. Greubel, Aug 10 2019

Formula

a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/5^k in lowest terms.
r(n) = Sum_{k=0..n} ((2*k-1)!!/((2*k)!!))*(4/5)^k, n>=0, with the double factorials A001147 and A000165.

A124396 Denominators of partial sums of a series for 3/sqrt(5) = (3/5)*sqrt(5).

Original entry on oeis.org

1, 9, 27, 729, 6561, 6561, 177147, 1594323, 4782969, 387420489, 3486784401, 10460353203, 282429536481, 2541865828329, 2541865828329, 22876792454961, 205891132094649, 617673396283947, 50031545098999707, 450283905890997363
Offset: 0

Views

Author

Wolfdieter Lang, Nov 10 2006

Keywords

Comments

Denominators of sums over central binomial coefficients scaled by powers of 9.
Numerators are given by A123749.
For the rationals r(n) see the W. Lang link under A123749.
This is not 3/5 times the rational sequence A123747/A123748 which converges to sqrt(5).

Examples

			a(3) = 729 because r(3) = 1 + 2/9 + 2/27 + 20/729 = 965/729 = A123749(3)/a(3).

Links

Crossrefs

Cf. A123749 (numerators).
Cf. A123747/A123748 partial sums for a series for sqrt(5).

Programs

  • GAP
    List([0..20], n-> DenominatorRat(Sum([0..n], k-> Binomial(2*k, k)/9^k)) ); # G. C. Greubel, Dec 25 2019
  • Magma
    [Denominator(&+[(k+1)*Catalan(k)/9^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019
  • Maple
    seq(denom(add(binomial(2*k, k)/9^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019
  • Mathematica
    Table[Denominator[Sum[(k+1)*CatalanNumber[k]/9^k, {k,0,n}]], {n,0,20}] (* G. C. Greubel, Dec 25 2019 *)
  • PARI
    a(n) = denominator(sum(k=0, n, binomial(2*k,k)/9^k)); \\ Michel Marcus, Aug 12 2019
  • Sage
    [denominator(sum((k+1)*catalan_number(k)/9^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

Formula

a(n) = denominator(r(n)) with the rationals r(n) = Sum_{k=0..n} binomial(2*k,k)/9^k in lowest terms.
r(n) = Sum_{k=0..n} ((2*k-1)!!/(2*k)!!)*(4/9)^k, n>=0, with the double factorials A001147 and A000165.

A124397 Numerators of partial sums of a series for sqrt(5)/3.

Original entry on oeis.org

1, 3, 21, 17, 99, 2223, 12039, 56763, 59337, 286961, 7358781, 36088473, 183146521, 181066401, 36534213, 4535753121, 22798981683, 113528187171, 113891192583, 568042152363, 14228623114839, 71035463999307, 355598139789279, 14210752102407, 1777633916948199
Offset: 0

Views

Author

Wolfdieter Lang, Nov 10 2006

Keywords

Comments

Denominators are given by A124398.
The alternating sums over central binomial coefficients scaled by powers of 5, r(n) = Sum_{k=0..n} (-1)^k*binomial(2*k,k)/5^k, have the limit s = lim_{n-> infinity} r(n) = sqrt(5)/3. From the expansion of 1/sqrt(1+x) for x=4/5.

Examples

			a(3) = 17 because r(3) = 1 - 2/5 + 6/25 - 4/25 = 17/25 = a(3)/A124398(3).

Links

Crossrefs

Cf. A123747/A123748 partial sums for a series for sqrt(5).
Cf. A123749/A124396 partial sums for a series for 3/sqrt(5).
Cf. A124398 (denominators), A208899 (sqrt(5)/3).

Programs

  • GAP
    List([0..20], n-> NumeratorRat(Sum([0..n], k-> (-1)^k*Binomial(2*k, k)/5^k)) ); # G. C. Greubel, Dec 25 2019
  • Magma
    [Numerator(&+[(-1)^k*(k+1)*Catalan(k)/5^k: k in [0..n]]): n in [0..20]]; // G. C. Greubel, Dec 25 2019
  • Maple
    seq(numer(add((-1)^k*binomial(2*k, k)/5^k, k = 0..n)), n = 0..20); # G. C. Greubel, Dec 25 2019
  • Mathematica
    Table[Numerator[Sum[(-1)^k*(k+1)*CatalanNumber[k]/5^k, {k,0,n}]], {n,0,20}] (* G. C. Greubel, Dec 25 2019 *)
  • PARI
    a(n) = numerator(sum(k=0, n, ((-1)^k)*binomial(2*k,k)/5^k)); \\ Michel Marcus, Aug 11 2019
  • Sage
    [numerator(sum((-1)^k*(k+1)*catalan_number(k)/5^k for k in (0..n))) for n in (0..20)] # G. C. Greubel, Dec 25 2019

Formula

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k * binomial(2*k,k)/5^k in lowest terms.
r(n) = Sum_{k=0..n} (-1)^k*((2*k-1)!!/(2*k)!!)*(4/5)^k, n>=0, with the double factorials A001147 and A000165.
Showing 1-4 of 4 results.

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